Proof of a symmetric group on a finite set not being cyclic if the set
has more than 2 elements. Doctor Carter debugs the work of a novice
algebraist's approach, which relies on non-commutative n-cycles; and
provides some pointers on conventions of notation.

Let G be a finite group. Show that there exists a positive integer "m"
such that a^m = e for all a in G. Suppose that G is a set closed under an
associative operation such that: for every a,y in G, there exists an x in
G such that ax = y; and for every a,w in G, there exsits a u in G such
that ua = w. Show that G is a group.

Let p be an odd prime. First, find a set of generators for a p-Sylow
subgroup K of S_p^2 (the symmetric group with degree p^2). Then find
the order of K and determine whether it is normal in S_p^2 and if it
is Abelian.